A power supply using a piezoelectric transformer is such a power supply that generates voltage using a resonance circuit. The piezoelectric transformer is several times larger than that of the conventional magnetic transformer in power density and can be operated in such a range of high frequency where the conventional magnetic transformer shows large loss and becomes impractical. The piezoelectric transformer can be made several times smaller in size than that of the magnetic transformer of the same rating in power. The piezoelectric transformer depends mainly on load and frequency, showing charac-teristics different from the conventional magnetic transformer, which prevents practical usage of the piezoelectric transformer in a power supply.
Such a stabilized direct current (dc) voltage supply regulates its output voltage using frequency dependence of amplitude ratio. The amplitude ratio, defined by the volt-age ratio of the input to the output of the piezoelectric transformer, shows resonance characteristics against the frequency of the carrier. From a viewpoint of efficiency, the piezoelectric transformer is supplied with such the carrier that is higher in frequency than the resonance of the piezoelectric transformer. For example, the frequency of the carrier is lowered and moved to the resonance to increase the output voltage. Yet the fall of the frequency causes the drop of the output voltage which is generated by rectifying the output carrier of the transformer. Namely, to lower the frequency in order to increase the output voltage causes temporary drop of the output voltage.
In the case that the carrier is higher than the resonance in frequency in the voltage supply, the transfer function of the output voltage has a zero in the right half plane as the function of the carrier frequency. It is difficult to implement a large loop gain for the circuit whose transfer function has zeros in the right half plane because of a narrow range of parameters for stable operation. Furthermore, it is difficult to resolve the zeros in the right half plane by feedback because the circuit having poles in the right half plane is not stable.
[Patent Citation 1]
Japanese Examined Patent Application Publication No. 4053255
[Patent Citation 2]
Japanese Examined Patent Application Publication No. 4268013
[Patent Citation 3]
Japanese Unexamined Patent Application Publication No. 2007-330091
[Patent Citation 4]
Japanese Unexamined Patent Application Publication No. 2008-306775
[Patent Citation 5]
PCT/JP2007/000477 WO2007129468
The patent reference 1 makes it a subject to offer the simple circuit of the dc power supply providing the stabilized high voltage with sufficient efficiency. Efficiency is improved by using not a conventional electromagnetic transformer but a piezoelectric transformer. The high voltage is stabilized by using the frequency dependability of the resonance characteristics of a piezoelectric transformer. The piezoelectric transformer simplifies the circuit and reduces parts in number, by which the subject is solved.
The patent reference 2 realizes improvement of the output voltage both in the accuracy of stabilization and in the speed of response by implementing feedback of a little delay, together with a large delay where the output voltage is stabilized based on the frequency dependability of the resonance characteristic. The present invention is that an idea of feedback in the patent reference 2 is applied to the amplitude of the carrier.
The patent reference 3 concerns the stabilized dc voltage power supply where stabilization is based on the frequency dependence of the resonance, giving composition and constants of the feedback circuit stabilizing the dc output voltage which is supplied to the load of a wide range. The transfer function feeding the output voltage back to the frequency of the carrier driving the resonance circuit has a pole located at the origin.
The patent reference 4 concerns the stabilized dc voltage power supply where stabilization is based on the frequency dependence of the resonance, giving composition and constants of the feedback circuit stabilizing the dc output voltage that is supplied to the load of a wide range. The transfer function feeding the output voltage back to the frequency of the carrier that drives the resonance circuit is not provided with the pole located at the origin.
The patent reference 5 is the PCT application based on the patent references 3 and 4.
The patent reference 3, the patent reference 4 and the patent reference 5 are surveyed with relation to the present invention. These patent references are concerned with the stabilized dc power supply generating the voltage with a resonance circuit.
Equivalent Power Supply Approximating DC Power Supply
A power supply providing the stabilized dc output voltage consists of a voltage generation circuit and the feedback circuit. The voltage generation circuit is composed of a driver circuit, the resonance circuit, and the rectification circuit. The driver circuit generates a high-frequency alternating current carrier of locally constant amplitude. The carrier drives the resonance circuit. The feedback circuit consists of an error amplifier and a voltage controlled oscillator (VCO). The error amplifier compares the dc output voltage with the reference voltage supplied externally to set the output voltage. The VCO is enabled to control the frequency of the carrier generated by the driver circuit. The output voltage of the power supply, which is the output of the rectification circuit, is fed back to the frequency of the carrier through the feedback circuit so as to be stabilized.
An equivalent power supply was developed in the patent references 3, 4 and 5, where the equivalent power supply approximates the dc stabilized power supply and can be analyzed by mathematical methods. The equivalent power supply is schematically composed of a virtual voltage generation circuit and a feedback circuit. The virtual voltage generation circuit includes a driver circuit generating a carrier of locally fixed amplitude, a virtual resonance circuit driven by the carrier supplied by the driver circuit and a virtual rectification circuit generating dc voltage from the output of the resonance circuit. The output of the power supply is the output of the rectification circuit. The feedback circuit includes an error amplifier comparing the output voltage with the reference voltage supplied externally to set the output voltage and a voltage controlled oscillator (VCO) generating the frequency decided by the output of the error amplifier. The voltage controlled oscillator is enabled to control the frequency of the carrier generated by the driver circuit, and thus the output voltage is fed back to the frequency of the carrier so as to be stabilized.
The virtual voltage generation circuit consists of a driver circuit, a virtual resonance circuit, and a virtual rectification circuit. The resonance circuit converts the modulation of the carrier from the frequency modulation at the input to the amplitude modulation at the output. The virtual resonance circuit, supplied with the carrier of the frequency modulation as is the case with the resonance circuit, outputs the envelope of the carrier modulated in amplitude, being different from the resonance circuit. The virtual rectification circuit inputs the envelope, acts as a filter of first-order delay to the envelope and outputs an output equivalent to the output of the rectification circuit.
Operation of the equivalent power supply is described by a system of differential equations, stability of which can be analyzed mathematically. The system of differential equations is derived, and then the necessary conditions that the output voltage of the equivalent power supply is stable in the neighborhood of the reference voltage are shown. An actual circuit, which realizes stable feedback based on the necessary conditions, is shown with circuit constants given explicitly.
Frequency Modulation, Imaginary Resonance and Rectification Circuits
As for the resonance circuit whose transfer function given by h, letting ωr, Q, and gr, be the angular velocity of the resonance frequency, the Q-value, and the amplitude ratio at the resonance frequency respectively of the resonance, then δ, ω0, and c are defined by
                    δ        =                              ω            r                                2            ⁢            Q                                              [        1        ]                                          ω          0                =                              ω            r                    ⁢                                    1              -                              1                                  4                  ⁢                                      Q                    2                                                                                                          [        2        ]                                c        =                                            g              r                        ⁢                          ω              r                                Q                                    [        3        ]            and the resonance circuit is driven by such a carrier of fixed amplitude modulated in frequency that is defined byωexp(iω0t+iψ)where ω is the amplitude of the carrier and ψ is a function of time representing a shift of phase, then
                    ϕ        =                              ⅆ                          ⅆ              t                                ⁢          ψ                                    [        5        ]            then the frequency of the carrier given in expression 4 isω0+φand, letting rr and ri be defined by
                              r          r                =                              1            2                    ⁢          c          ⁢                                          ⁢          w                                    [        7        ]                                          r          i                =                              δ                          2              ⁢                              ω                0                                              ⁢          c          ⁢                                          ⁢          w                                    [        8        ]            the resonance circuit, being driven by the carrier in expression 4, outputs the carrier the amplitude of which is given by p and q as√{square root over (p2+q2)}where p and q satisfy
                                          ⅆ                          ⅆ              t                                ⁢          p                =                              q            ⁢                                                  ⁢            ϕ                    -                      p            ⁢                                                  ⁢            δ                    +                      r            r                                              [        10        ]                                                      ⅆ                          ⅆ              t                                ⁢          q                =                                            -              p                        ⁢                                                  ⁢            ϕ                    -                      q            ⁢                                                  ⁢            δ                    +                      r            i                                              [        11        ]            
Then a first order delay which, letting the dc voltage generated by rectifying the carrier outputted by the resonance circuit be z, is represented by the following differential equation concerning z
                                          μ            ⁢                          ⅆ                              ⅆ                t                                      ⁢            z                    +          z                =                  ν          ⁢                                                    p                2                            +                              q                2                                                                        [        12        ]            where μ and ν are a time constant and a multiplier at the rectification circuit respectively.Feedback Circuit and System of Differential Equations
Letting k, d, E, A, and B be positive numbers and λ be a reference voltage respectively, the dc voltage z in expression 12 from the rectification circuit is compared with the reference voltage λ. The voltage difference between z and λ is fed back to the frequency of the carrier φ in expression 5, where the feedback is expressed on the assumption that φ>0 by the transfer function having a pole located at the origin as
                    ϕ        =                  k          ⁢                                          ⁢          d          ⁢                                    (                              E                +                                  A                  ⁢                                                                          ⁢                  s                                +                                  B                  ⁢                                                                          ⁢                                      s                    2                                                              )                        s                    ⁢                      (                          z              -              λ                        )                                              [        13        ]            
Then uniting expression 13, expressions 10, expression 11, and expression 12 makes the system of differential equations describing the power supply as
                                              ⁢                                            ⅆ                              ⅆ                t                                      ⁢            p                    =                                    q              ⁢                                                          ⁢              ϕ                        -                          p              ⁢                                                          ⁢              δ                        +                          r              r                                                          [        14        ]                                                          ⁢                                            ⅆ                              ⅆ                t                                      ⁢            q                    =                                                    -                p                            ⁢                                                          ⁢              ϕ                        -                          q              ⁢                                                          ⁢              δ                        +                          r              i                                                          [        15        ]                                                          ⁢                                            ⅆ                              ⅆ                t                                      ⁢            z                    =                                                    -                z                            +                              ν                ⁢                                                                            p                      2                                        +                                          q                      2                                                                                            μ                                              [        16        ]                                                      ⅆ                          ⅆ              t                                ⁢          ϕ                =                              k            ⁢                                                  ⁢            E            ⁢                                                  ⁢                          d              ⁡                              (                                  z                  -                  λ                                )                                              +                                    k              ⁢                                                          ⁢              A              ⁢                                                          ⁢                              d                ⁡                                  (                                                            -                      z                                        +                                          ν                      ⁢                                                                                                    p                            2                                                    +                                                      q                            2                                                                                                                                )                                                      μ                    +                      k            ⁢                                                  ⁢            B            ⁢                                                  ⁢                          d              ⁡                              (                                                      -                                                                                                                                                      p                              2                                                        +                                                          q                              2                                                                                                      ⁢                                                  (                                                      ν                            +                            νμδ                                                    )                                                                                            μ                        2                                                                              +                                      z                                          μ                      2                                                        +                                                            ν                      ⁡                                              (                                                                              q                            ⁢                                                                                                                  ⁢                                                          r                              i                                                                                +                                                      p                            ⁢                                                                                                                  ⁢                                                          r                              r                                                                                                      )                                                                                                                                                                  p                            2                                                    +                                                      q                            2                                                                                              ⁢                      μ                                                                      )                                                                        [        17        ]            Stability and Overshoots
The stability of feedback in the stabilized dc power supply is attributed to the stability of the system of the differential equations given by expressions 14˜17. The stability of the system is decided by the root of the characteristic polynomial derived from the system of the differential equations. That all the roots of the characteristic polynomial have negative real parts is the necessary and sufficient condition that the system of differential equations is stable in the sense of Lyapunov. It is the necessary condition to be satisfied by a power supply that the system of differential equations describing the power supply is stable in the sense of Lyapunov. Yet stability in the sense of Lyapunov is not enough for the stability of the power supply. For instance, there exist the cases where the power supply stable in the sense of Lyapunov oscillates the output voltage. There are also cases where the output voltage oscillates in the neighborhood of the voltage set by the reference voltage and cases where the oscillation decays in a long time.
There is operation of the power supply which the system of differential equations given by the expressions 14˜17 does not approximate. The output voltage of the power supply is generated by the rectification circuit. The rectification circuit includes a capacitor, and the output voltage is buffered by the capacitor. As for the positive output voltage of the rectification circuit, pumping up the charge to the capacitor raises the output voltage, while it is impossible to pumping out the charge from the capacitor so as to lower the output voltage. Then the raise of the output voltage can be described by the system of the differential equations 14˜17, while there are falls of the output voltage which the system of the differential equations cannot reproduce correctly.
While the output voltage is higher than the reference voltage, the feedback works to lower the output voltage, reducing the current supplied to the rectification circuit by the resonance circuit. In the case that the current is reduced to zero while the output voltage is higher than the reference voltage, the output voltage falls with a time constant of load resistance and the capacitor in the rectification circuit, where the fall is independent of the system of the differential equations. When the output voltage becomes lower than the reference voltage, the feedback begins to work, raising the output voltage. If the rise of the output voltage is accompanied by overshoots in voltage, the output voltage begins repeating the rise and the fall in voltage. Then it is necessary for stable feedback that the rise of the output voltage is free from the overshoot.
A Sufficient Condition
In the case that the characteristic polynomial has the real characteristic root separated from complex roots, the rise of the output voltage is not accompanied by the overshoot and then the feedback is stable. Let the transfer function feeding the output voltage back to the frequency of the carrier be given by
                              (                                    E              s                        +            1            +                          B              ⁢                                                          ⁢              s                                )                ×        N                            [        18        ]            where N is a positive constant, then 1/E is an approximation of the time constant where the output voltage is raised. Letting E and B be assigned as
                    E        <                  1          μ                                    [        19        ]                        and                                                      B        ∼                  1          δ                                    [        20        ]            the bandwidth for feedback be restricted by the time constant 1/E, and N be assigned so that the loop gain becomes enough smaller than unity at the zero in the right half plane, then the feedback becomes stable.Zeros in Right Half Plane
A zero in the right half plane is created, for example, by the resonance circuit driven by the carrier which is higher in frequency than the resonance frequency of the resonance circuit. In the stabilized dc power supply where the output voltage is generated by rectifying the output of the resonance circuit and stabilized by feeding the output voltage back to the frequency of the carrier supplied to the resonance circuit, the output voltage is lowered by increasing and raised by lowering the frequency of the carrier in the case that a frequency range of the carrier is selected to be higher than the resonance frequency of the resonance circuit as shown in FIG. 1. The fall of the frequency causes the drop of the output voltage. Namely, to lower the frequency in order to increase the output voltage causes temporary drop of the output voltage.
When the frequency of the carrier is lowered and moved to the resonance to raise the output voltage, the fall of the frequency causes immediate drop of the output voltage. The amplitude of the carrier outputted by the resonance circuit is changed after the frequency is shifted. The time delay from the shift of the frequency to the change of the amplitude is approximated by 1/δ. The amplitude of the temporary drop is dependent of magnitude of the load. The control which makes the output voltage increase accompanies the temporal drop of the output voltage, which is characteristic of the control system provided with zeros located at the right half plane.